Estimation efficiency in a binary mixed-effects model setting

We investigate the efficiency of likelihood methods for estimating the regression parameters of mixed-effects logistic regression models. One approach uses a conditional likelihood which eliminates the random intercept terms. A second uses the likelihood generated from the marginal distribution of the data where the random intercepts are integrated out. Parametric estimates result from assuming a parametric form for the intercept distribution, whereas we obtain semiparametric estimates when the intercept distribution is left unspecified. We present an expression which shows that the asymptotic relative efficiency of conditional likelihood estimators relative to parametric estimators is a decreasing function of within-cluster covariate correlation. Simulation results show the same for the asymptotic relative efficiency of the semiparametric estimator. relative to the conditional. For fixed covariate correlation, the asymptotic relative efficiency of the parametric versus the conditional increases as cluster sizes increase. Example data further illustrate our findings.